Page - 278 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 andthen ∂g ∂α (β,λ)= 1 2 ∫ T [ϕ−1(q)−ϕ−1(p)]ϕ′(cβ+λu0)dμ. (20) Forβ∈ (α0,α1)andλ∈ (0,κ0), the function inside the integral sign in (20) isdominatedby f. Asaresult,aseconduseof theDominatedConvergenceTheoremshowsthat ∂g∂α iscontinuousat (α,κ): lim (β,λ)→(α,κ) ∂g ∂α (β,λ)= ∂g ∂α (α,κ). Usingsimilararguments,onecanshowthat ∂g∂κ(α,κ)existsandiscontinuousatanyα∈ (α0,α1) andκ>0,andisgivenby ∂g ∂κ (α,κ)= ∫ T u0ϕ′(cα+κu0)dμ. (21) Clearly,expression(21) implies that ∂g∂κ(α,κ)>0 forallα∈ (0,α0)andκ>0. Weprovedthat items(i)–(iii) aresatisfied.Asconsequence, thederivativeofκ(α)existsatany α∈ (α0,α1). Expression(16) for thederivativeofκ(α) followsfrom(17), (20)and(21). Asan immediateconsequenceofLemma2,weget thepropositionbelow. Proposition1. Assumethat ϕ(·) is continuouslydifferentiable. (a) If, for someα0<−1, expression (15) is satisfied forallα∈ [α0,−1), then D(−1)ϕ (p‖ q)= lim α↓−1 D(α)ϕ (p‖ q)=2∂κ∂α(−1)=Dϕ(p‖ q)<∞. (b) If, for someα1>1, expression (15) is satisfied forallα∈ (1,α1], then D(1)ϕ (p‖ q)= lim α↑1 D(α)ϕ (p‖ q)=−2∂κ∂α(1)=Dϕ(q‖ p)<∞. 4.GeneralizedStatisticalManifolds Statisticalmanifoldsconsistofacollectionofprobabilitydistributionsendowedwithametric andα-connections,whicharedefinedin termsof thederivativeof l(t;θ)= logp(t;θ). Inageneralized statisticalmanifold, themetricandconnectionaredefinedintermsof f(t;θ)= ϕ−1(p(t;θ)). Instead of the logarithm,weconsider the inverse ϕ−1(·)of a ϕ-function. Generalizedstatisticalmanifolds were introducedbytheauthors in [17,18]. Amongexamplesof thegeneralizedstatisticalmanifold, (parametric)ϕ-familiesofprobabilitydistributionsareofgreatest importance. Thenon-parametric counterpartwas investigated in [11,12]. Themetric inϕ-familiescanbedefinedas theHessianofa function; i.e., ϕ-familiesareHessianmanifolds [30]. In [17,18], the ϕ-divergencegives rise toapair ofdualconnectionsD(−1) andD(1); andthenforα∈ (−1,1) theα-connectionD(α) isdefinedas the convexcombinationD(α) = 1−α2 D (−1)+ 1+α2 D (1). In thepresentpaper,weshowthat theconnection inducedbyD(α)ϕ (· ‖ ·), thegeneralizationofRényidivergence, corresponds toD(α). 4.1.Definitions Let ϕ:R→ (0,∞)bea ϕ-function. Ageneralized statisticalmanifoldP= {p(t;θ) : θ ∈Θ} is a collectionofprobabilitydistributions pθ(t) := p(t;θ), indexedbyparametersθ=(θ1, . . . ,θn)∈Θ in aone-to-onerelation, suchthat (m1) Θ isadomain(openandconnectedset) inRn; (m2) p(t;θ) isdifferentiablewithrespect toθ; 278